This is a tricky one. I find that if kids learn to sing at a young age, they can then sing forever. It's like, in order to create notes, you have to hear it, then your brain has to process it, then ask the muscles in your throat to contract or relax in order to form the correct tension to produce that note.
I have found that if you never use these "muscles" then the brain finds it very hard to engage with the system, and no connection is made. I am SURE it is a brain thing. I have also noticed that many kids who can't pitch are dyslexic- which makes me more convinced that it is a brain thing rather than just lack of practice or training.
Tell your student to feel the tension in her throat on high and low notes. When I teach kids harmony singing and they have to pitch a difficult harmony out of thin air, I tell them to sing the note and "feel" where it is in their throat, in their range. Very often they think it is lower and therefore more comfortable, so don't pitch high enough, and end up singing an octave below the tune which always sounds horrible!
I agree though that there is only a certain amount you can do, and it may be better to suggest she learns an instrument "to accompany her singing" which will make her a better musician and listener- piano or guitar. THEN deal with the singing a bit later.
I have never come across a 6 year old who is tone deaf, but I have come across several teenagers who are- and this is because they have never used their ears or stretched their (vocal) muscles, and got the brain linked up with it all. It is very hard to go from this level to be singing in tune once you are a teenager.. She is never going to be Christina or Mariah, sadly.